Remark

The first axiom guarantees that we have a functor J:Cop→SetJ: C^{op} \to Set. Thus JJ itself can be regarded as an object of the presheaf topos[Cop,Set][C^{op},Set]; in this way Grothendieck topologies on CC are identified with Lawvere-Tierney topologies on [Cop,Set][C^{op},Set].

Given a Grothendieck topology JJ on a small categoryCC, one can define the category Sh(C,J)Sh(C,J) of sheaves on CC relative to JJ, which is a reflective subcategory of the category [Cop,Set][C^{op},Set] of presheaves on CC. Thus we have a functor C→Sh(C,J)C\to Sh(C,J) given by the composite of the Yoneda embedding with the reflection (or “sheafification”). This composite functor is fully faithful if and only if all representable presheaves are sheaves for JJ; a topology with this property is called subcanonical.

Saturation

Grothendieck topologies may be and in practice quite often are obtained as closures of collections of morphisms that are not yet closed under the operations above (that are not yet sieves, not yet pullback stable, etc.).

Two notions of such unsaturated collections of morphisms inducing Grothendieck topologies are

Examples

Topology on open subsets of a topological space

The archetypical example of a Grothendieck topology is that on a category of open subsetsOp(X)Op(X) of a topological spaceXX. A covering family of an open subset U⊂XU \subset X is a collection of open subsets Vi⊂UV_i \subset U that cover UU in the ordinary sense of the word, i.e. which are such that every point x∈Ux \in U is in at least one of the ViV_i.

Canonical topology

On any category there is a largest subcanonical topology. This is called the canonical topology, with “subcanonical” a back-formation from this (since a topology is subcanonical iff it is contained in the canonical topology). On a Grothendieck topos, the covering families in the canonical topology are those which are jointly epimorphic.

Related notions

A more general notion is simply a collection of “covering families,” not necessarily sieves, satisfying only pullback-stability; this suffices to define an equivalent notion of sheaf. Following the Elephant, we call such a system a coverage. A Grothendieck topology may then be defined as a coverage that consists of sieves (which the Elephant calls “sifted”) and satisfies certain extra saturation conditions; see coverage for details.

An intermediate notion is that of a Grothendieck pretopology, which consists of covering families that satisfy some, but not all, of the closure conditions for a Grothendieck topology. Many examples are “naturally” pretopologies, but must be “saturated” under the remaining closure conditions to produce Grothendieck topologies.